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Multiplying Whole Numbers

How to Multiply Two Whole Numbers to Find the Product: Write the numbers so each place value lines up vertically. Multiply the digits in each place value. Work from right to left, starting with the ones place in the bottom number. Multiply the bottom ...

Multiplying Whole Numbers

How to Multiply Two Whole Numbers to Find the Product:

  1. Write the numbers so each place value lines up vertically.
  2. Multiply the digits in each place value.
    1. Work from right to left, starting with the ones place in the bottom number.
      • Multiply the bottom number by the ones digit in the top number, then by the tens digit, and so on.
      • If a product in a place value is more than \(9,\) carry to the next place value.
      • Write the partial products, lining up the digits in the place values with the numbers above.
    2. Repeat for the tens place in the bottom number, the hundreds place, and so on.
    3. Insert a zero as a placeholder with each additional partial product.
  3. Add the partial products.

Example

Multiply: \(62\left(87\right).\)

Solution

Write the numbers so each place lines up vertically. CNX_BMath_Figure_01_04_020_img-02.png
Start by multiplying 7 by 62. Multiply 7 by the digit in the ones place of 62. \(7\cdot 2=14.\) Write the 4 in the ones place of the product and carry the 1 to the tens place. CNX_BMath_Figure_01_04_020_img-03.png
Multiply 7 by the digit in the tens place of 62. \(7\cdot 6=42.\) Add the 1 ten we carried. \(42+1=43\). Write the 3 in the tens place of the product and the 4 in the hundreds place. CNX_BMath_Figure_01_04_020_img-04.png
The first partial product is 434.
Now, write a 0 under the 4 in the ones place of the next partial product as a placeholder since we now multiply the digit in the tens place of 87 by 62. Multiply 8 by the digit in the ones place of 62. \(8\cdot 2=16.\) Write the 6 in the next place of the product, which is the tens place. Carry the 1 to the tens place. CNX_BMath_Figure_01_04_020_img-05.png
Multiply 8 by 6, the digit in the tens place of 62, then add the 1 ten we carried to get 49. Write the 9 in the hundreds place of the product and the 4 in the thousands place. CNX_BMath_Figure_01_04_020_img-06.png
The second partial product is 4960. Add the partial products. CNX_BMath_Figure_01_04_020_img-07.png

The product is \(5,394.\)

Extra:

Multiply: \(43\left(78\right).\)

Solution

3,354

Extra:

Multiply: \(64\left(59\right).\)

Solution

3,776

Example

Multiply:

  1. \(47·10\)
  2. \(47·100.\)

Solution

1. \(47·10\). \(\begin{array}{r} 47\\ ×10\\ \hline 00\\ 470\\ \hline 470\end{array}\)
2. \(47·100\) \(\begin{array}{r} 47\\ ×100\\ \hline 00\\ 000\\ 4700\\ \hline 4,700\end{array}\)

When we multiplied \(47\) times \(10,\) the product was \(470.\) Notice that \(10\) has one zero, and we put one zero after \(47\) to get the product. When we multiplied \(47\) times \(100,\) the product was \(4,700.\) Notice that \(100\) has two zeros and we put two zeros after \(47\) to get the product.

Do you see the pattern? If we multiplied \(47\) times \(10,000,\) which has four zeros, we would put four zeros after \(47\) to get the product \(470,000.\)

Extra:

Multiply:

  1. \(54·10\)
  2. \(54·100.\)

Solution

  1. 540
  2. 5,400

Extra:

Multiply:

  1. \(75·10\)
  2. \(75·100.\)

Solution

  1. 750
  2. 7,500

Example

Multiply: \(\left(354\right)\left(438\right).\)

Solution

There are three digits in the factors so there will be \(3\) partial products. We do not have to write the \(0\) as a placeholder as long as we write each partial product in the correct place.

An image of the multiplication problem “354 times 438” worked out vertically. 354 is the top number, 438 is the second number. Below 438 is a multiplication bar. Below the bar is the number 2,832. 2832 has the label “Multiply 8 times 354”. Below 2832 is the number 1,062; 1062 has the label “Multiply 3 times 354”. Below 1062 is the number 1,416; 1416 has the label “Multiply 4 times 354”. Below this is a bar and below the bar is the number “155,052”, with the label “Add the partial products”.

Extra:

Multiply: \(\left(265\right)\left(483\right).\)

Answer

127,995

Extra:

Multiply: \(\left(823\right)\left(794\right).\)

Answer

653,462

Example

Multiply: \(\left(896\right)201.\)

Solution

There should be \(3\) partial products. The second partial product will be the result of multiplying \(896\) by \(0.\)

An image of the multiplication problem “896 times 201” worked out vertically. 896 is the top number, the 8 in the hundreds place, the 9 in the tens place, the 6 in the ones place. 201 is the second number, the 2 in the hundreds place, the 0 in the tens place, the 1 in the ones place. Below 201 is a multiplcation bar. Below the bar is the number 896, the 8 in the hundreds place, the 9 in the tens place, the 6 in the ones place. 896 has the label “Multiply 1 times 896”. Below 896 is the number “000”, the 0 in the thousands place, the 0 in the hundreds place, and the 0 in the tens place. “000” has the label “Multiply 0 times 896”. Below “000” is the number 1792, the 1 in the hundred thousands place, the 7 in the ten thousands place, the 9 in the thousands place, and the 2 in the hundreds place. 1792 has the label “Multiply 2 times 896”. Below this is a bar and below the bar is the number “180,096”, with the label “Add the partial products”.

Notice that the second partial product of all zeros doesn’t really affect the result. We can place a zero as a placeholder in the tens place and then proceed directly to multiplying by the \(2\) in the hundreds place, as shown.

Multiply by \(10,\) but insert only one zero as a placeholder in the tens place. Multiply by \(200,\) putting the \(2\) from the \(12.\) \(2·6=12\) in the hundreds place.

\(\begin{array}{r}\hfill 896\phantom{\rule{0.5em}{0ex}}\\ \hfill \underset{\text{_____}}{×201}\phantom{\rule{0.3em}{0ex}}\\ \hfill 896\phantom{\rule{0.5em}{0ex}}\\ \hfill \underset{\text{__________}}{17920}\phantom{\rule{0.29em}{0ex}}\\ \hfill 180,096\phantom{\rule{0.5em}{0ex}}\end{array}\)

Extra:

Multiply: \(\left(718\right)509.\)

Answer

365,462

Extra:

Multiply: \(\left(627\right)804.\)

Answer

504,108

When there are three or more factors, we multiply the first two and then multiply their product by the next factor. For example:

to multiply \(8\cdot 3\cdot 2\)
first multiply \(8\cdot 3\) \(24\cdot 2\)
then multiply \(24\cdot 2\). \(48\)

Optional Video: Example on Multiplying Whole Numbers

This video below by Mathispower4u provides an example of multiplying a 3-digit whole number and a 2-digit whole number.

This lesson is part of:

Introducing Numbers

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